# Nonlinear approximations

Linear model reduction methods rely on the introduction of a linear subspace to approximate the solution field over a range of parameters. Linear methods are effective to devise efficient reduced-order models (ROMs) for a broad class of problems in mechanics; however, the inadequacy of linear approximations to deal with parametric fields with sharp gradients fundamentally hinders the application of linear model reduction to advection-dominated problems with shocks or wakes. This motivates the interest towards nonlinear approximations. In our team, we develop approaches based on domain decomposition and registration methods.

## Domain decomposition

We develop a domain decomposition method to couple high-dimensional and reduced-order models. In this approach, we use the HDM to solve the model where a given degree of accuracy is required, while the ROM is used to approximate the solution elsewhere. Since the HDM is used in a small part of the domain, the computational cost can be significantly reduced.

We evaluate the performance of this domain decomposition method in 1D and 2D. We first consider a supersonic flow in a converging-diverging nozzle. The domain is divided in three parts as shown by Figure 1.

Figure 1: In parts 1 and 3, ROMs are used to approximate the flow and the HDM is used in part 2 to accurately modelled the shock wave.

In Figure 2, we give an example of prediction test. The error of the resulting model is lower than 1% and the run time is reduced by approximately 73%.

Figure 2: The travelling shock wave is accurately approximated.

Then we consider a transonic flow over a NACA 0012 airfoil. The domain is divided in two parts as shown by Figure 3.

Figure 3: The HDM is used inside the black circle to accurately model the shock wave and the strong gradients, while the ROM is used elsewhere.

An example of prediction test is shown in Figure 4. The error of the resulting model is lower than 1% and the run time is reduced by approximately 78%.

## Registration methods

In computer vision and pattern recognition, registration refers to the process of finding a spatial transformation that aligns two datasets; in the framework of model reduction, registration aims to find a parametric transformation (i.e., a bijective mapping) to track moving features such as shocks and contact discontinuities, and ultimately improve performance of linear compression methods — such as proper orthogonal decomposition.

To demonstrate the potential of registration, consider the problem of approximating the space-time solution to the shallow water equations for a parametric inflow condition (see here for the full details on this test case). Linear approximations are extremely inaccurate and fail to recover the behaviour of the solution, especially in the proximity of the shock; on the other hand, the registration-based approximation offers accurate out-of-sample reconstructions, for the same number of modes.

linear MOR (projection for N=5)

registration-based MOR (prediction for N=5)

Registration is also important to reduce the size of the required mesh to achieve a certain threshold: in order to achieve accurate results, mesh should be refined in the proximity of sharp structures for all parameter values. By tracking moving features in the reference configuration, registration provides a framework to perform parametric r-adaptivity and ultimately improve the accuracy of the high-fidelity discretization for a given mesh budget.